Let a(x) and b(x) – b.m. functions at x® a (x® + ¥, x® –¥, x® x 0 , …). Consider the limit of their ratio at x® a.

1. If = b and b- final number b¹ 0, then the functions a(x), b(x) are called infinitesimal one order of magnitude at x® a.

2. If = 0, then a(x) is called infinitesimal higher order , how b(x) at x® a. Obviously, in this case = ¥.

3. If a(x) – b.m. higher order than b(x), and = b¹ 0 ( b- final number kÎ N ), then a(x) is called infinitesimal k-th order compared to b(x) at x® a.

4. If does not exist (neither finite nor infinite), then a(x), b(x) are called incomparable b.m. at x® a.

5. If = 1, then a(x), b(x) are called equivalent b.m. at x® a, which is denoted as follows: a(x) ~ b(x) at x® a.

Example 1. a(x) = (1 – x) 3 , b (x) = 1 – x 3 .

It is obvious that at x® 1 functions a(x), b(x) are b.m. To compare them, we find the limit of their ratio at x® 1:

Conclusion: a(x b(x) at x® 1.

It is easy to verify that = (make sure!), whence it follows that a(x) – b.m. 3rd order of smallness, compared to b(x) at x® 1.

Example 2. Functions a 1 (x) = 4x, a 2 (x) = x 2 , a 3 (x) = sin x, a 4 (x) = tg x are infinitesimal for x® 0. Compare them:

0, , = 1, = ¥.

Hence we conclude that a 2 (x) = x 2 - b.m. higher order than a 1 (x) and a 3 (x) (at x® 0), a 1 (x) and a 3 (x) – b.m. one order, a 3 (x) and a 4 (x) are equivalent b.m., i.e. sin x~tg x at x® 0.

Theorem 1. Let a(x) ~ a 1 (x), b(x) ~ b 1 (x) at x® a. If exists , then exists and , and = .

Proof. = 1, = 1,

= = .

This theorem makes it easier to find the limits.

Example 3.

Find .

By virtue of the first remarkable limit of sin4 x~ 4x, tg3 x~ 3x at x® 0, so

Theorem 2. Infinitely small functions a(x) and b(x) are equivalent (for x® a) if and only if a(x) – b(x) is b.m. higher order than a(x) and b(x) (at x® a).

Proof

Let a(x) ~ b(x) at x® a. Then = = 0, i.e. difference a(x) – b(x a(x) at at x® a(similar to b(x)).

Let a(x) – b(x) – b.m. higher order than a(x) and b(x), we will show that a(x) ~ b(x) at x® a:

= = + = 1,

Infinitely small functions.

We continue the training cycle "limits for dummies", which opened with articles Limits. Solution examples and Remarkable Limits. If this is your first time on the site, I recommend that you also read the lesson Limit Solving Methods which will greatly improve your student karma. In the third manual, we considered infinite functions, their comparison, and now it is time to arm yourself with a magnifying glass, so that after the Land of the Giants, look into the Land of the Lilliputians. I spent the New Year holidays in the cultural capital and returned to a very good mood, so reading promises to be especially interesting.

This article will discuss in detail infinitesimal functions, with which you have actually already encountered many times, and their comparison. Many events are closely related to invisible events near zero. wonderful limits, wonderful equivalences, and the practical part of the lesson is mainly devoted to just calculating the limits using wonderful equivalences.

Infinitely small functions. Comparison of infinitesimals

What can I say ... If there is a limit, then the function is called infinitesimal at a point.

The essential point of the assertion is the fact that function can be infinitesimal only at a specific point .

Let's draw a familiar line:

This function infinitely small at a single point:
It should be noted that, at the points "plus infinity" and "minus infinity", the same function will be already infinitely large: . Or in a more compact notation:

At all other points, the limit of the function will be equal to a finite number other than zero.

In this way, there is no such thing as "just an infinitely small function" or "just an infinitely large function". A function can be infinitesimal or infinitely large only at a specific point .

! Note : for brevity, I will often say "infinitesimal function", meaning that it is infinitely small at the point in question.

There can be several or even infinitely many such points. Let's draw some kind of fearless parabola:

The presented quadratic function is infinitely small at two points - at "one" and at "two":

As in the previous example, at infinity, this function is infinitely large:

The meaning of double signs :

The notation means that at , and at .

The notation means that both at , and at .
The commented principle of "deciphering" double signs is valid not only for infinities, but also for any end points, functions and a number of other mathematical objects.

And now the sine. This is an example where the function infinitely small at an infinite number of points:

Indeed, the sinusoid "flashes" the x-axis through each "pi":

Note that the function is bounded from above/below, and there is no such point at which it would be infinitely large, the sine can only lick its lips at infinity.

Let me answer a couple of simple questions:

Can a function be infinitesimal at infinity?

Of course. Such instances of a cart and a small cart.
Elementary example: . The geometric meaning of this limit, by the way, is illustrated in the article Graphs and properties of functions.

Can a function NOT be infinitesimal?
(at any point domains)

Yes. An obvious example is a quadratic function whose graph (parabola) does not intersect the axis. The converse statement, by the way, is generally not true - the hyperbole from the previous question, although it does not cross the x-axis, but infinitely small at infinity.

Comparison of infinitesimal functions

Let's build a sequence that tends to zero, and calculate several values ​​of the trinomial:

It is obvious that with a decrease in the x values, the function runs away to zero faster than all the others (its values ​​are circled in red). They say that a function than a function , as well as higher order of smallness, how . But running fast in the Land of the Lilliputians is not valor, the “tone is set” by the slowest dwarf, who, as befits the boss, goes to zero the slowest of all. It depends on him how fast the sum will approach zero:

Figuratively speaking, an infinitely small function “absorbs” everything else, which is especially well seen in the final result of the third line. Sometimes they say that lower order of smallness, how and their sum.

In the considered limit, all this, of course, does not really matter, because the result is still zero. However, the “heavyweight midgets” begin to play on principle important role within fractions. Let's start with examples that, although rare, are found in real life. practical work:

Example 1

Calculate Limit

There is uncertainty here, and introductory lesson about functions we recall the general principle of disclosing this uncertainty: you need to decompose the numerator and denominator into factors, and then reduce something:

At the first step, we take out the brackets in the numerator, and "x" in the denominator. In the second step, we reduce the numerator and denominator by "x", thereby eliminating the uncertainty. We indicate that the remaining "X's" tend to zero, and we get the answer.

In the limit, the bagel turned out, therefore, the numerator function higher order of smallness than the denominator function. Or shorter: . What does it mean? The numerator tends to zero faster than the denominator, which is why the result is zero.

As in the case with infinite functions, the answer can be known in advance. The reception is similar, but differs in that in the numerator and in the denominator you need to MENTALLY discard all the terms with SENIOR degrees, since, as noted above, slow dwarfs are of decisive importance:

Example 2

Calculate Limit

Zero to zero…. Let's find out the answer right away: MENTALLY discard everything elder terms (fast dwarfs) of the numerator and denominator:

The solution algorithm is exactly the same as in the previous example:

In this example denominator of a higher order of smallness than the numerator. When the x values ​​decrease, the slowest dwarf of the numerator (and the entire limit) becomes a real monster in relation to its faster opponent. For example, if , then - already 40 times more .... not yet a monster, of course, with the given value of "x", but such is already a subject with a big beer belly.

And a very simple demo limit:

Example 3

Calculate Limit

We will find out the answer by MENTALLY discarding everything elder numerator and denominator terms:

We decide:

The result is a finite number. The owner of the numerator is exactly twice as thick as the boss of the denominator. This is the situation where the numerator and denominator one order of magnitude.

In fact, the comparison of infinitesimal functions has long appeared in previous lessons:
(Example number 4 lesson Limits. Solution examples);
(Example No. 17 lesson Limit Solving Methods) etc.

I remind you at the same time that "x" can tend not only to zero, but also to an arbitrary number, as well as to infinity.

What is fundamentally important in all the examples considered?

Firstly, the limit must exist at all at a given point. For example, there is no limit. If , then the numerator function is not defined at the point "plus infinity" (under the root we get infinitely large a negative number). Similar, it would seem, pretentious examples are found in practice:, no matter how unexpected, here is also a comparison of infinitesimal functions and the uncertainty "zero to zero". Indeed, if , then . …Solution? We get rid of the four-story fraction, get the uncertainty and open it with the standard method.

Perhaps, beginners to explore the limits are drilled by the question: “How so? There is an uncertainty of 0:0, but you cannot divide by zero! Quite right, you can't. Let's consider the same limit. The function is not defined at point "zero". But this, generally speaking, is not required. important for the function to exist IN ANY infinitely close to zero point (or more strictly, at any infinitesimal neighborhood zero).

THE MOST IMPORTANT FEATURE OF THE LIMIT AS A CONCEPT

is that "x" infinitely close approaches a certain point, but he is “not obliged to go there”! That is, for the existence of a function limit at a point irrelevant whether the function itself is defined there or not. You can read more about this in the article. Cauchy limits, but for now, back to the topic of today's lesson:

Secondly, the numerator and denominator functions must be infinitesimal at a given point. So, for example, the limit is from a completely different team, here the numerator function does not tend to zero: .

We systematize the information on comparing infinitesimal functions:

Let - infinitesimal functions at a point(i.e. at ) and there is a limit of their ratios . Then:

1) If , then the function higher order of smallness, how .
The simplest example: , that is, a cubic function of a higher order of smallness than a quadratic one.

2) If , then the function higher order of smallness, how .
The simplest example: , that is, a quadratic function of a higher order of smallness than a linear one.

3) If , where is a nonzero constant, then the functions have same order of magnitude.
The simplest example: , in other words, the dwarf runs to zero strictly two times slower than , and the "distance" between them remains constant.

The most interesting case is when . Such functions are called infinitesimal equivalent functions.

Before giving an elementary example, let's talk about the term itself. Equivalence. This word has already been used in class. Limit Solving Methods, in other articles and will meet more than once. What is equivalence? There is a mathematical definition of equivalence, logical, physical, etc., but let's try to understand the essence itself.

Equivalence is equivalence (or equivalence) in some respect. It's time to stretch your muscles and take a break from higher mathematics. It is now a good January frost outside, so it is very important to warm up well. Please go to the hallway and open the closet with clothes. Imagine that there are two identical sheepskin coats hanging there, which differ only in color. One is orange, the other is purple. In terms of their warming qualities, these sheepskin coats are equivalent. Both in the first and in the second sheepskin coat you will be equally warm, that is, the choice is equivalent to what to wear orange, what purple - without winning: "one to one is equal to one." But from the point of view of safety on the road, sheepskin coats are no longer equivalent - the orange color is better visible to drivers of transport, ... and the patrol will not stop, because everything is clear with the owner of such clothes. In this regard, we can assume that sheepskin coats of “one order of smallness”, relatively speaking, “orange sheepskin coat” are twice as “safer” than “purple sheepskin coats” (“which is worse, but also noticeable in the dark”). And if you go out into the cold in one jacket and socks, then the difference will be already colossal, thus, a jacket and a sheepskin coat are “of a different order of smallness”.

… zashib, you need to post on Wikipedia with a link to this lesson =) =) =)

The obvious example of infinitely small equivalent functions is familiar to you - these are the functions first remarkable limit .

Let us give a geometric interpretation of the first remarkable limit. Let's execute the drawing:

Well, the strong male friendship of the graphs is visible even to the naked eye. BUT their own mother will not distinguish them. Thus, if , then the functions are infinitesimal and equivalent. What if the difference is negligible? Then in the limit the sine above can be replace"x": , or "x" below the sine: . In fact, it turned out to be a geometric proof of the first remarkable limit =)

Similarly, by the way, one can illustrate any wonderful limit, which is equal to one.

! Attention! Object equivalence does not imply the same objects! Orange and purple sheepskin coats are equivalent to warm, but they are different sheepskin coats. The functions are practically indistinguishable near zero, but they are two different functions.

Designation: equivalence is indicated by a tilde.
For example: - "the sine of x is equivalent to x", if .

A very important conclusion follows from the above: if two infinitesimal functions are equivalent, then one can be replaced by the other. This technique is widely used in practice, and right now we will see how:

Remarkable equivalences within

To solve practical examples, you will need remarkable equivalence table. The student does not live as a single polynomial, so the field of further activity will be very wide. First, using the theory of infinitesimal equivalent functions, we recap the examples of the first part of the lesson Remarkable limits. Solution examples, in which the following limits were found:

1) Let's solve the limit. Let's replace the infinitesimal function of the numerator with the equivalent infinitesimal function :

Why is this substitution possible? because infinitely close to zero the graph of the function almost coincides with the graph of the function.

In this example, we used table equivalence where . It is convenient that not only “x”, but also a complex function can act as the “alpha” parameter, which tends to zero.

2) Let's find the limit. In the denominator we use the same equivalence , in this case :

Please note that the sine was originally under the square, so in the first step it is also necessary to place it entirely under the square.

Do not forget about the theory: in the first two examples, finite numbers are obtained, which means that numerators and denominators of the same order of smallness.

3) Find the limit. Let us replace the infinitesimal function of the numerator with the equivalent function , where :

Here numerator of a higher order of smallness than the denominator. Lilliput (and its equivalent midget) reaches zero faster than .

4) Find the limit. Let us replace the infinitely small function of the numerator with an equivalent function , where :

And here, on the contrary, the denominator higher order of smallness than the numerator, the dwarf runs away to zero faster than the dwarf (and its equivalent dwarf).

Should wonderful equivalences be used in practice? It should, but not always. Thus, the solution of not very complex limits (like those just considered) is undesirable to be solved through remarkable equivalences. You can be reproached for hack-work and forced to solve them in a standard way using trigonometric formulas and the first wonderful limit. However, with the help of the tool in question, it is very beneficial to check the solution or even immediately find out the correct answer. Characteristic Example No. 14 of the lesson Limit Solving Methods:

On a clean copy, it is advisable to draw up a rather large complete solution with a change of variable. But the ready answer lies on the surface - we mentally use the equivalence: .

Once again geometric meaning: why in the numerator it is permissible to replace the function with the function ? Infinitely close to zero their graphs can only be distinguished under a powerful microscope.

In addition to checking the solution, wonderful equivalences are used in two more cases:

– when the example is rather complicated or even undecidable in the usual way;
– when remarkable equivalences need to be applied by condition.

Let's consider more meaningful tasks:

Example 4

Find the limit

Zero-to-zero uncertainty is on the agenda and the situation is borderline: a decision can be made in a standard way, but there will be many transformations. From my point of view, it is quite appropriate to use the wonderful equivalences here:

Let us replace infinitesimal functions with equivalent functions. At :

That's all!

The only technical nuance: initially the tangent was squared, so after the replacement, the argument must also be squared.

Example 5

Find the limit

This limit can be solved through trigonometric formulas and wonderful limits, but the solution will again not be very pleasant. This is an example for self-solving, be especially careful during the conversion of the numerator. If there is confusion with powers, represent it as a product:

Example 6

Find the limit

But this is already a difficult case, when it is very difficult to carry out a solution in a standard way. We use wonderful equivalences:

Let us replace infinitesimal ones with equivalent ones. At :

Infinity is obtained, which means that the denominator is of a higher order of smallness than the numerator.

The practice went briskly without outerwear =)

Example 7

Find the limit

This is a do-it-yourself example. Think about how to deal with the logarithm ;-)

It is not uncommon to see remarkable equivalences used in combination with other limit-solving methods:

Example 8

Find the limit of a function using equivalent infinitesimals and other transformations

Note that the remarkable conditional equivalences need to be applied here.

We decide:

At the first step, we use remarkable equivalences. At :

Everything is clear with the sine: . What to do with the logarithm? We represent the logarithm in the form and apply the equivalence . As you can see, in this case

In the second step, we apply the technique discussed in the lesson

What are infinite small functions

However, a function can be infinitely small only at a specific point. As shown in Figure 1, the function is infinitesimal only at point 0.

Figure 1. An infinitesimal function

If the quotient limit of two functions results in 1, the functions are said to be equivalent infinitesimal as x approaches a.

\[\mathop(\lim )\limits_(x\to a) \frac(f(x))(g(x)) =1\]

Definition

If the functions f(x), g(x) are infinitesimal for $x > a$, then:

• The function f(x) is called an infinitesimal higher order with respect to g(x) if the following condition is satisfied:
\[\mathop(\lim )\limits_(x\to a) \frac(f(x))(g(x)) =0\]
• The function f(x) is called infinitesimal of order n with respect to g(x) if it is different from 0 and the limit is finite:
\[\mathop(\lim )\limits_(x\to a) \frac(f(x))(g^(n) (x)) =A\]

Example 1

The function $y=x^3$ is an infinitesimal higher order for x>0, in comparison with the function y=5x, since the limit of their ratio is 0, this is explained by the fact that the function $y=x^3$ tends to zero value faster:

\[\mathop(\lim )\limits_(x\to 0) \frac(x^(2) )(5x) =\frac(1)(5) \mathop(\lim )\limits_(x\to 0 )x=0\]

Example 2

The functions y=x2-4 and y=x2-5x+6 are infinitesimal of the same order for x>2, since the limit of their ratio is not equal to 0:

\[\mathop(\lim )\limits_(x\to 2) \frac(x^(2) -4)(x^(2) -5x+6) =\mathop(\lim )\limits_(x\ to 2) \frac((x-2)(x+2))((x-2)(x-3)) =\mathop(\lim )\limits_(x\to 2) \frac((x+ 2))((x-3)) =\frac(4)(-1) =-4\ne 0\]

Properties of equivalent infinitesimals

1. The difference of two equivalent infinitesimals is an infinitesimal of higher order with respect to each of them.
2. If we discard infinitesimal higher orders from the sum of several infinitesimal different orders, then the remaining part, called the main part, is equivalent to the entire sum.

It follows from the first property that equivalent infinitesimals can become approximately equal with an arbitrarily small relative error. Therefore, the sign ≈ is used both to denote the equivalence of infinitesimals and to write the approximate equality of their sufficiently small values.

When finding limits, it is very often necessary to use a change of equivalent functions for the speed and convenience of calculations. The table of equivalent infinitesimals is presented below (Table 1).

The equivalence of the infinitesimals given in the table can be proved based on the equality:

\[\mathop(\lim )\limits_(x\to a) \frac(f(x))(g(x)) =1\]

Table 1

Example 3

Let us prove the equivalence of infinitesimal ln(1+x) and x.

Proof:

1. Find the limit of the ratio of quantities
\[\mathop(\lim )\limits_(x\to a) \frac(\ln (1+x))(x) \]
2. To do this, we use the property of the logarithm:
\[\frac(\ln (1+x))(x) =\frac(1)(x) \ln (1+x)=\ln (1+x)^(\frac(1)(x) ) \] \[\mathop(\lim )\limits_(x\to a) \frac(\ln (1+x))(x) =\mathop(\lim )\limits_(x\to a) \ln (1+x)^(\frac(1)(x) ) \]
3. Knowing that the logarithmic function is continuous in its domain of definition, you can swap the sign of the limit and the logarithmic function:
\[\mathop(\lim )\limits_(x\to a) \frac(\ln (1+x))(x) =\mathop(\lim )\limits_(x\to a) \ln (1+ x)^(\frac(1)(x) ) =\ln \left(\mathop(\lim )\limits_(x\to a) (1+x)^(\frac(1)(x) ) \ right)\]
4. Since x is an infinitesimal value, the limit tends to 0. So:
\[\mathop(\lim )\limits_(x\to a) \frac(\ln (1+x))(x) =\mathop(\lim )\limits_(x\to a) \ln (1+ x)^(\frac(1)(x) ) =\ln \left(\mathop(\lim )\limits_(x\to 0) (1+x)^(\frac(1)(x) ) \ right)=\ln e=1\]

(applied the second remarkable limit)

Test

Discipline: Higher mathematics

Subject: Limits. Comparison of infinitesimals

1. Limit of number sequence

2. Function limit

3. Second remarkable limit

4. Comparison of infinitesimal quantities

Literature

1. Limit of number sequence

The solution of many mathematical and applied problems leads to a sequence of numbers given in a certain way. Let's find out some of their properties.

Definition 1.1. If every natural number

according to some law, a real number is put in correspondence, then the set of numbers is called a numerical sequence.

Based on Definition 1, it is clear that a numerical sequence always contains an infinite number of elements. The study of various numerical sequences shows that as the number increases, their members behave differently. They can increase or decrease indefinitely, they can constantly approach a certain number or not show any regularity at all.

Definition 1.2. Number

is called the limit of a numerical sequence if for any number there is such a number of a numerical sequence depending on that the condition is satisfied for all numbers of the numerical sequence.

A sequence that has a limit is called convergent. In this case, write

.

Obviously, to clarify the question of the convergence of a numerical sequence, it is necessary to have a criterion that would be based only on the properties of its elements.

Theorem 1.1.(Cauchy's theorem on the convergence of a numerical sequence). For a numerical sequence to converge, it is necessary and sufficient that for any number

there was such a numerical sequence number depending on that for any two numbers of the numerical sequence and that satisfy the condition and , the inequality would be true.

Proof. Need. It is given that the numerical sequence

converges, which means that, according to Definition 2, it has a limit . Let's pick some number. Then, according to the definition of the limit of a numerical sequence, there is such a sequence number , that for all numbers the inequality is fulfilled. But since it is arbitrary, it will be fulfilled and . Let's take two sequence numbers and , then .

Hence it follows that

, that is, the necessity is proven.

Adequacy. Given that

. Hence, there exists a number such that for the given condition and . In particular, if , and , then or provided that . This means that the numerical sequence for is limited. Therefore, at least one of its subsequences must converge. Let . Let us prove that converges to also.

Let's take an arbitrary

. Then, according to the definition of the limit, there exists a number such that the inequality holds for all . On the other hand, by condition it is given that the sequence has such a number that for all and the condition will be satisfied. and fix some . Then for all we get: .

Hence it follows that